Nonadiabatic Ab Initio Multichannel Quantum Defect Theory Applied to Absolute Experimental Absorption Intensities in H2

Краткое содержание

Multichannel quantum defect theory (MQDT) combined with
the frame transformation concept is perhaps the most successful
theoretical approach capable of treating electronically excited
molecular systems beyond the Bo
-Oppenheimer approximation.
1-4 It surpasses the traditional coupled equations approach
in that it is not restricted to just a few excited states
which have to be added to the treatment one by one but instead
handles whole families of excited states (Rydberg series) up to
the ionization thresholds and beyond. Furthermore, using the
powerful concept of frame transformations, MQDT bypasses
the evaluation of the electronic coupling matrix elements, which
require the knowledge of numerical ab initio electronic wave
functions. A quantitative comparison of the two approaches may
be made by referring to two papers which appeared more than
10 years ago,5,6 where each theory was applied to the same
vibronic problem.
Numerous applications of MQDT to molecular spectroscopic
problems have been published during the last decades, progressing
from rotationally resolved absorption,4,7 emission,8 or
multiphoton ionization spectroscopy and dynamics9,10 to hyperfine
resolved millimeter wave and photoionization spectroscopy.
11-13 However, to our knowledge, no quantitative MQDT
study of molecular line intensities of any molecule has been
made so far in a situation where absolute measured line
intensities were available for comparison.
In this paper, we present a fully ab initio calculation of the
line positions and intensities of the Q(N) (N ) 1-4) absorption
transitions in molecular hydrogen H2, based on MQDT. We
extract the quantum defects and channel dipole transition
moments from the latest ab initio calculations of Wolniewicz
and collaborators14-16 and use them in the framework of the
nonadiabatic frame transformation-MQDT. We obtain overall
good agreement with the recent synchrotron-based absolute
intensity measurements of Glass-Maujean et al.17 It is wellknown
that the energy positions of the excited levels of H2 of
1Πu
- symmetry (which are the upper states reached by the Q
transitions) are only little affected by vibronic interactions. By
contrast, the intensities of the Q transitions are surprisingly
strongly perturbed by the 1Πu
- ∼ 1Πu
- vibronic interactions in
many instances, even when the line positions are apparently
unperturbed. This fact had been established experimentally in
ref 17 and is confirmed by the present calculations.
2. Determination of Quantum Defects and Channel
Transition Moments from Quantum Chemical Data
We use quantum defect theory in its simplest form by
disregarding channel interactions between singly excited and
doubly (core) excited Rydberg channels. In other words, we
assume that the manifold of the 1Πu excited states of H2
represent a single unperturbed npπ Rydberg series converging
to the X+ 2Σg
+ ground state of H2
+. Correlation between the
excited electron and the ion core electron are included in an
effective manner in the quantum defects because we extract the
latter from highly accurate theoretical clamped nuclei (Bo
-
Oppenheimer) potential energy curves14,16 in which electron
correlation has been fully accounted for. We use the familiar
one-channel Rydberg equation
written here in atomic energy units in order to extract the
quantum defects for 1Πu symmetry. Unpπ(R) is the clamped
nuclei curve for the npπ Rydberg state and U +(R) is the
corresponding curve for the ion ground state, X+ 2Σg
+. We used
the data for n ) 2 (C state), 3 (D state), and 4 (D′ state) from
ref 14 (see Figure 1), which yield the quantum defect curves
for n ) 2, 3, and 4. The set of clamped nuclei quantum defect
† Part of the “Robert W. Field Festschrift”.
* To whom correspondence should be addressed.
‡ Universite´ Pierre et Marie Curie.
§ Universite´ de Paris-Sud.
⊥ University College London.
Unpπ(R) ) U+(R) - 1
2[n - μnpπ(R)]2 (1)
J. Phys. Chem. A XXXX, xxx, 000 A
10.1021/jp902846c CCC: $40.75  XXXX American Chemical Society
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Published on June 23, 2009 on http://pubs.acs.org | doi: 10.1021/jp902846c
curves thus obtained is subsequently represented by an energydependent
polynomial of the form
where
( R) ) Unpπ(R) - U+(R) is the binding energy of the
Rydberg electron in the field of the core with the nuclei kept
fixed at a distance R. (The fourth term on the rhs of eq 2 is
discussed below.) The process of reducing the three available
ab initio curves Unpπ (R) to the three functions μ(0)(R), μ(1)(R),
and μ(2)(R) is illustrated in Figure 1a-c. Note that the functions
μ(0)(R), μ(1)(R), and μ(2)(R) allow us to recover the potential
energy curves for n ) 2-4 exactly (because for each R, three
values have been represented in terms of three parameters) by
means of eqs 1 and 2, but we also can predict all of the higher
curves with n > 4. The reliability of this extrapolation (within
reasonable limits) is suggested by the fact that the three quantum
defect curves in Figure 1b nearly coincide.
The last term in eq 2 corresponds to the “specific” mass effect
(mass polarization term) arising from the cross term H3′ )
-(m/4M)∇1∇2 in the molecular Hamiltonian (where m is the
electron mass and M is the nuclear reduced mass). This term
couples the Rydberg and the core electrons and, as shown in
ref 4, contributes a small mass-dependent correction to the
quantum defect. In quantum chemical computations, the same
term arises as part of the adiabatic corrections but unfortunately
is rarely given separately when these are evaluated. Following
ref 4, we take it here independent of R, and energy. We use the
value μspecific ) -0.16 for the npπ1Πu channel.
A similar procedure is applied to the ab initio dipole transition
moments from refs 14 and 16. We first reduce the values
DXfnpπ(R) (Figure 1d) to energy-normalized transition moments
dXfnpπ(R) by multiplying by the square root of the density of
states dν/d
) (n - μnpπ)3/2 (in au, where ν ) n - μ ) (-2
)-1/2)
(Figure 1e). It may be seen that the energy-normalized moments
d, while exhibiting less energy dependence than the moments
D shown at the top, are by no means as constant as their
quantum defect counterparts represented in Figure 1b. The
reasons for this will be discussed in section 6 below. We note
on the other hand that the photon absorption from the ground
state effectively takes place in a small region centered around
the equilibrium position near 1.4 a0, where the energy dependence
of the d’s is relatively mild. We finally represent these
reduced moments for each R by an expression analogous to
eq 2
The resulting channel dipole transition functions d(0)(R), d(1)(R),
and d(2)(R) for excitation from the ground state to the pπ
Rydberg channel are shown in Figure 1f. In agreement with
what has just been said, the quantities d(1) and d(2) are seen to
be rather small in the region around Re ) 1.4 a0.